Problem Statement of Series 1, Year 26

Try to find out dimensions, mass and density of a regular A4 paper (one that you would use in a printer). Given these values estimate its average thickness.

Estimate the number of electrons in an adult human body.

Imagine that there are wires along all lines of latitude and longitude. These wires are connected at each point where they cross. Given that the resistance of one meter of this wire is $\rho$, calculate the resistance of this network between the nodes located at the North and South pole. Assume that the angular distance of neighboring lines of latitude/longitude is 15$°$.

Bonus: What would be the resistance between two nodes located on the equator and opposite of each other?

Two cars are driving towards each other with speed $v_0$. At what distance from each other should they hit the brakes in order to avoid a crash? Consider the case when both cars are on a flat road as well as the case when they are on a road inclined at an angle $\alpha$. Both drivers apply the brakes at the same time. The braking force is equal to $f\cdot N$ where $N$ is the component of the car's weight normal to the road.

Consider the classical Double-slit experiment but assume that the screen behind the two slits is cylindrical. The axis of this cylinder is parallel with the the two slits and the whole apparatus is symmetrical. This axis is at a distance $L$ from the slits, radius of the cylindrical screen is $R=L⁄2$ and the separation of the two slits is $a$. Describe the diffraction pattern on the screen after it has been unrolled from the cylinder and give the locations of diffraction maxima using a coordinate corresponding to the distance on the cylinder's surface.

What is the minimal wind speed necessary to blow away a paper lying on a flat table?

In this problem you are asked to investigate some elastic properties of a human hair. First you should carry out an experiment to obtain the dependence of the magnitude of the applied force on the change of the hair's length. This will allow you to construct a graph describing the dependence of the applied stress on the strain. Using these data try to estimate the maximal allowed stress and perhaps also other characteristics of the hair. You should make at least 3 measurements using hair from the same person.

Hint: The longer the hair, the better. You can measure the diameter of a hair using a micrometer or a laser in your school. Convenient weights are coins which usually have a well defined mass.

• Find out typical properties of a plasma present is the solar wind, in the center of a tokamak and in a low-pressure discharge and calculate the corresponding value of ${\lambda }_D$.

• Derive an expression for the Debye length of plasma that consists of electrons at temperature $T_e$ and ions at temperature $T_i$. Do not assume that the ions are static.

• Calculate the electrostatic potential of two infinite parallel conducting planes that are separated by a distance $d$. The potential on these planes is kept $\phi =0$ and the space in between these planes is filled with a gas of particles with charge $q$ and concentration $n$.